8.
WHAT KATZ ACTUALLY SAID “we assume that each link independently has the same probability of being effective” … “we conceive a constant , depending on the group and the context of the particular investigation, which has the force of a probability of effectiveness of a single link. A k-step chain then, has probability of being effective.” “We wish to find the column sums of the matrix” Leo Katz 1953, A New Status Index Derived from Sociometric Analysis, Psychometria 18(1):39-43David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 8 of 47

11.
Carl Neumann I’ve heard the Neumann series called the “von Neumann” series more than I’d like! In fact, the von Neumann kernel of a graph should be named the “Neumann” kernel! Wikipedia pageDavid F. Gleich (Purdue) Univ. Chicago SSCS Seminar 11 / 47

16.
WHAT DO OTHER PEOPLE DO?1) Just work with the linear algebra formulations2) For Katz, Truncate the Neumann series as a few (3-5) terms3) Use low-rank approximations from EVD(A) or EVD(L)4) For commute, use Johnson-Lindenstrauss inspired random sampling5) Approximately decompose into smaller problems Liben-Nowell and Kleinberg CIKM2003, Acar et al. ICDM2009, Spielman and Srivastava STOC2008, Sarkar and Moore UAI2007,Wang et al. ICDM2007David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 16 of 47

17.
THE PROBLEM All of these techniques are preprocessing based because most people’s goal is to compute all the scores. We want to avoid preprocessing the graph. There are a few caveats here! i.e. one could solve the system instead of looking for the matrix inverseDavid F. Gleich (Purdue) Univ. Chicago SSCS Seminar 17 of 47

25.
MMQ PROCEDUREGoal Given 1. Run k-steps of Lanczos on starting with 2. Compute , with an additional eigenvalue at , set Correspond to a Gauss-Radau rule, with u as a prescribed node3. Compute , with an additional eigenvalue at , set Correspond to a Gauss-Radau rule, with l as a prescribed node4. Output as lower and upper bounds on David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 25 of 47

40.
CONVERGENCE?If 1/max-degree then is sub-stochastic and the PageRank based proofapplies because the matrix is diagonallydominantFor , then for symmetric ,this algorithm is the Gauss-Southwellprocedure and it still converges.David F. Gleich (Purdue) Univ. Chicago SSCS Seminar 40 of 47